https://file.aiquickdraw.com/mathbot/file/1727922604135sk4yjozl.jpg

The function presented in the image is:

\[
f(t) = t^6 e^t
\]

To find the derivative \( f'(t) \), we will use the **product rule** because this function is the product of two differentiable functions: \( t^6 \) and \( e^t \).

The product rule states that if \( f(t) = g(t) \cdot h(t) \), then:

\[
f'(t) = g'(t) \cdot h(t) + g(t) \cdot h'(t)
\]

Here, let:

- \( g(t) = t^6 \), so \( g'(t) = 6t^5 \)
- \( h(t) = e^t \), so \( h'(t) = e^t \) (since the derivative of \( e^t \) is just \( e^t \))

Applying the product rule:

\[
f'(t) = g'(t) \cdot h(t) + g(t) \cdot h'(t)
\]

\[
f'(t) = 6t^5 \cdot e^t + t^6 \cdot e^t
\]

This simplifies to:

\[
f'(t) = e^t(6t^5 + t^6)
\]

Thus, the derivative of the function is:

\[
f'(t) = e^t(t^6 + 6t^5)
\]

Would you like more details on any of these steps or further clarification?

Here are 5 related questions to expand your understanding:
1. How is the product rule applied to other functions?
2. How would you differentiate a function like \( t^n e^{kt} \) (with a constant \( k \))?
3. What other rules (like the chain rule) could be involved in more complex derivatives?
4. How can you use derivatives to find the slope of the tangent line to a curve at a given point?
5. Can you apply the quotient rule to functions that involve division?

**Tip:** Always look for cases where you can factor out common terms when simplifying derivatives!

Find the derivative of the following function: f(t) = t^6 * e^t

Learn how to find the derivative of the function f(t) = t^6 * e^t using the product rule. This calculus problem involves differentiating a product of two functions, t^6 and e^t. Ideal for students in Grades 11-12 or college-level calculus.

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